When the vertex angle and the base of an isosceles triangle are given how do you find its area?

Let's A and x represent the given vertex angle and the base, respectively.

Use the law of cosine to find the length of the legs of the triangle by doing x2 = m2 + n2 - 2mncos A, where m and n are the legs. Since the triangle is isosceles, m = n and therefore x2 = 2m2 - 2m2cos A. Solving for m gives m = sqrt(x2/(2 - cos A))

Get the height of the triangle by using Pythagorean theorem. m2 = x2 + h2, where h is the height.

Finally, get the area using the formula for a triangle's area, which is (base * height) / 2.